Sunday, September 07, 2014

This Is *Not* The Best Way To Learn Math

Watch this Common Core "how to add" video first:

Of course, there's nothing wrong with "deconstructing" the 6 so that you can get a 9+1=10. In fact, that's how I often add large numbers. But to say that there's something wrong with memorizing basic arithmetic tables up to 10+10? Let's remember what our friends at Stanford and the NIH have to say about that:

When it comes to adding up it's experience that counts, scientists have found.

Research carried out on elementary school-age children has revealed that drilling children on simple addition and multiplication may pay off.

According to the results, as children's brains develop remembering sums helps them add up faster.

'Experience really does matter,' said Dr Kathy Mann Koepke of the National Institutes of Health, which funded the research.

Healthy children start making that switch between counting to what's called fact retrieval when they're eight to nine-years-old, when they're still working on fundamental addition and subtraction.

How well children make that shift to memory-based problem-solving is known to predict their ultimate math achievement...

Next, Menon's team put 20 adolescents and 20 adults into the MRI machines and gave them the same simple addition problems. It turns out that adults don't use their memory-crunching hippocampus in the same way. Instead of using a lot of effort, retrieving six plus four equals 10 from long-term storage was almost automatic, Menon said.

In other words, over time the brain became increasingly efficient at retrieving facts. Think of it like a bumpy, grassy field, NIH's Mann Koepke explained.

Walk over the same spot enough and a smooth, grass-free path forms, making it easier to get from start to end.

If your brain doesn't have to work as hard on simple maths, it has more working memory free to process the teacher's brand-new lesson on more complex math.

In response to a petition signed by over 17,000 parents, the Alberta government included times table memorization in its curriculum this September. Manitoba added the same requirement one year ago but went a step further to include standard methods like long division and declared JUMP Math, developed by a Canadian charity, a recommended resource. The forward-thinking Winnipeg School Division, which is the largest division in Manitoba, will be adopting JUMP Math in many classrooms this fall. Last spring, Ontario Minister of Education Liz Sandals announced that children in Ontario should be required to memorize times tables, but the Ontario government has not taken formal action to ensure this.

Ontario’s recent EQAO results showed that the percentage of Grade 6 students who meet provincial standards fell from 61 to 54 per cent over the past five years. Education Minister Liz Sandals claimed that, contrary to public opinion, the EQAO test results revealed that students did not have difficulties with basic arithmetic but that problem solving stumped students. Sandals neglected to mention that Grade 6 students were permitted the use of calculators and manipulatives like blocks throughout the entire test. It is not surprising that students struggled with problem solving but the ability to push buttons on a calculator does not reflect fluency with basic arithmetic.

The Ontario government should look closely at the two textbook series used in most Ontario elementary schools – Pearson’s Math Makes Sense and Nelson Mathematics. Authors of these texts claim to nurture creative thinking, problem solving and understanding of math concepts. If this is true, why has student performance in math declined over the period in which these texts have been used? It is time to adopt alternatives that include less fuzzy instructional techniques like JUMP Math, Singapore Math or Saxon Math.

Solid education research that conclusively demonstrates the effectiveness of particular instructional techniques is hard to find. Nonetheless, over the last 10 years, teaching methods have tended towards discovery-based instruction, also referred to as inquiry-based learning, 21st century learning or constructivism. In this environment, explicit or direct instruction from teachers is minimized, rigorous practice and memorization of facts is discouraged, group work is the norm and novice learners are encouraged to invent their own strategies for solving open-ended math problems with little direction from adults.

Project Follow Through, which involved 72,000 students over a period of 10 years, starting in 1968 was the largest education study ever conducted. The study concluded that Direct Instruction, characterized by explicit instruction followed by practice, feedback and assessment, resulted in students who had better basic skills, better understanding of math concepts and better problem-solving skills and confidence than those taught using discovery techniques. Ironically, students educated using discovery techniques are less likely to be strong problem solvers than those educated using conventional techniques. Students need toolboxes stocked with knowledge, facts and well-practised skills in order to solve challenging problems and to understand math deeply.

Recent research in cognitive science confirms what Project Follow Through found forty years earlier. A 2011 meta analysis of 164 studies led by psychologist Louis Alfieri concluded that explicit instruction, worked examples and feedback benefit learners while unassisted discovery does not. A 2014 study published in Educational Evaluation and Policy Analysis found that only direct instruction, routine practice and drill significantly improved math achievement in struggling math students. Repetition and practice give students the vehicle for storing important knowledge and techniques into long-term memory so that they can be quickly accessed later. Another article, which appeared in Nature Neuroscience in August, found that failure to memorize math facts early results in impaired math learning later on. On the other hand, I have not found one rigorous study showing that discovery-based instruction is more effective than conventional instruction.

I've said it forever: it's taken the best minds the human race has had to offer thousands of years to discover and create the mathematics we have today. It's ridiculous to expect teenagers to figure it out for themselves in an hour a day.

10 comments:

Of course there's something wrong here. I'm all in favor of someone knowing why 9+6= 15, but the teacher just used the words 'partnering, ' and 'decomposing,' for use with a student who is barely learning to add... and, in the process, made what should be a problem that one can do in one's head into a lengthy, tiresome process ... you could achieve the same result with a few questions: "You've got 9, and we're adding 6 ... we want the total. How many would it take the 9 to get to ten? If we took them from the 6, how many would be left? " Done. Writing this out makes math become tedious and hated ...

This is the biggest crock I have seen since Whole Language. After the instructor got everyone "comfortable" and "deconstructed"part of the problem, she still had to rely on knowledge of addition combinations to arrive at the correct answer.

The idiotic straw man argument that in the past we were only taught to memorize the abstract symbols of 9 + 6 = 15, without understanding that it meant anything more than scribbles on a page needs to stop!

There has never been a curriculum that didn't start with representing the abstract numbers with either real-world physical counters or with diagrams on a page. Then moving towards understanding how you can add two piles together to get another number, as well as how you can group the objects together to make use of place value. Word problems which count students in a classroom or fruit in a bowl do the same thing.

Much of K, 1st, 2nd, and (when it comes to extending place value out into the quadrillions) even 3rd is taken up--and has always been taken up--by the tying of the real world to the abstract world of numbers on a page. Every curriculum and every standard I have ever seen for the early grades focuses on place value and understanding of arithmetic, not the memorization of abstract and meaningless symbols.

These twits that keep saying: in the past teachers and text book writers were unenlightened Gradgrinds needlessly torturing our children with meaningless memorization, but now we are blessed with the enlightened attitude that students actually need to understand what they are doing! need to get a clue.

Since Lehrer was making fun of something that looks an awful lot like the way I was taught (except for the base-8 stuff, though I did have to do some of that too,) and laughing at what I consider a fairly standard algorithm...I always wonder what way he thinks is better?

I think it's simple: things that require memorization, you require memorization ... but that doesn't mean you can't ALSO explain why it works ...except for long division. That's a very clever algorithm that no one really needs to understand. I never really understood why it worked, until I started teaching synthetic division ... and then, it just sort of clicked.